3. Overview of some recent results

Striving to offer a more general approach, the authors of article [11] did not confine ourselves to a specific process governing shifts, but allowed them to belong to a certain class of functions, such as, for example, polynomials of degree less than some specified number n (pp. 858–861). In this way, they did not expose themselves to any model misspecification risk, similarly as

The larger class of shifts (diseases) against which the immunization will work, the better. Having this in mind, the interval ½ � t0; T was divided in [6] into n equal nonoverlapping subintervals Ik, 1 ≤ k ≤ n, and set akðÞ¼ t 1 when t∈ Ik and akðÞ¼ t 0 otherwise (p. 34). The admissible

k¼m

k¼1

with A0ð Þt representing an instantaneous rate of cash payout, while sk standing for single payment at instances t1, t2, t3, …, tm: The expression δð Þ t � tk , with δð Þt standing for a Dirac delta function, was employed in order to make integration possible. The following result

Theorem 1. If q denotes the date when the single liability of L dollars has to be discharged by means of the cumulative value of assets A(t), then the immunization is secured against all

> A tð Þ exp ½ � �hð Þ 0; t t VA

where VA stands for the present value of the portfolio represented by assets A(t).

Remark 1. When n = 1, then Theorem 1 gives a sufficient and necessary condition

A tð Þ exp ½ � �hð Þ 0; t t VA

for immunization when the term structure h(0,t) is subject to shifts h tð Þþ ; 0 λa tð Þ. In case of

remembered as the following statement: Immunization is secured if the duration of the assets

Remark 2. Theorem 1 can also be looked at from a different perspective. Namely, one may be interested in identification of such a set of shifts a(t) of the term structure s(t) of interest rates,

ðT 0

λkakð Þt of interest rates h(0,t) if and only if

0

A tð Þ exp ½ � �hð Þ 0;t t

k P¼n k¼1

ckδð Þ t � tk , (6)

akð Þt tdt, 1 ≤ k ≤ n, (7)

a tð Þtdt (8)

VA tdt, which is well

λkakð Þt . The authors

shifts were assumed to be piecewise constant functions of the form

k P¼n k¼1

ðT 0

a qð Þq ¼

parallel shifts, (8) reduces to Fisher-Weil condition <sup>q</sup> <sup>¼</sup> <sup>Ð</sup> <sup>T</sup>

DA equals the duration of the single liability.

akð Þq q ¼

A tðÞ¼ <sup>A</sup>0ð Þþ <sup>t</sup> <sup>X</sup>

made a general assumption stating that a BP generates inflows

(Theorem 3, pp. 34–35 in [6]) was then proved.

100 Immunization - Vaccine Adjuvant Delivery System and Strategies

adverse piecewise constant shifts

Zheng in [12].

In a recent paper [13], the authors found a strong evidence that momentum across various asset classes is caused by macroeconomic variables. By properly modifying their portfolio, in response to changes in macroeconomic environment, their strategy performed particularly well in times of economic distress. The obtained results allowed them to establish a link between momentum and sophisticated predictive regressions.

Aiming at securing higher effectiveness of their investment in fixed income bonds, the authors of [14] successfully used simulations of the portfolio surplus, measuring the inherent risk by means of the value-at-risk methodology. In another very recent publication [15], the authors studied immunization assuming that shifts were parallel or symmetric. A quite different approach to immunization was proposed in [16]. The authors concentrated on hedging risk inherent in bond portfolio. They divided the entire problem into two parts, by formulating a two-step optimization problem. They focused first on immunization risk, and next maximized the portfolio wealth.

In this section, it is proved that the set of all continuous shocks a(t) against which a bond portfolio BP is immunized is an m-dimensional linear subspace in the (m + 1)-dimensional linear space of all continuous shifts a(t), with m standing for the number of instances when BP promises to pay cash (coupons or par values generated by bonds forming BP). The main mathematical concept used below is the notion of a Hilbert space and the concept of a base in a Hilbert space.

#### 3.1. When polynomials are admissible shifts

From now on, we assume that A0ðÞ� t 0 in Formula 6, so that inflows given by

$$A(t) = \sum\_{k=1}^{k=m} c\_k \delta(t - t\_k),\tag{9}$$

generate only payments ck at specified instances t1, t2, t3, …, tm. In such a situation, the present value of assets A(t) is no longer given by Eq. (2), but by

$$V\_A = \sum\_{k=1}^{k=m} c\_i \exp\left[-c(t\_k)t\_k\right]. \tag{10}$$

a qð Þ<sup>q</sup> <sup>¼</sup> <sup>X</sup> i¼m

> i P¼m i¼1

more, the subset of these polynomials satisfying Eq. (13) is a linear space, too.

Eq. (13) holding for c(t), one immediately obtains the required relationship.

½ � b qð Þþ c qð Þ <sup>q</sup> <sup>¼</sup> <sup>X</sup>

How to determine IMMU is demonstrated in [11] in pp. 858–860.

infinite number of independent vectors!

3.2. Continuous functions are admissible shifts: a Hilbert space approach

with weights

i P¼m i¼1

sion n.

i¼1

wk <sup>¼</sup> ck exp ½ � �s tð Þ<sup>k</sup> tk

Remark 4. The class of polynomials of the form (11), with fixed n, is a linear space. What is

Proof. Assume that a(t) satisfies Eq. (13). For any real number r, Eq. (13) implies ½ � ra qð Þ q ¼

i¼m

i¼1

Theorem 2. (see [11], Theorem 2.1). Let q denote the date when the single liability of L dollars has to be discharged by means of the cumulative value of assets (9) despite additive adverse shifts (11) (n is fixed) of interest rates s(t) so that the new interest rates will be of the form (12). Then, the subclass of shifts (11) for which immunization is secured is a (n�1)-dimensional linear space, denote it by IMMU, of the space of all polynomials (11), which itself has dimen-

The other class of admissible shifts studied in [11] was the class of all continuous functions (CF) defined as always on interval ½ � t0; T . As previously, the new interest rates (after a shift) satisfy Eq. (12) with a(t) standing this time for any CF. As previously, assets A(t) are given by Formula 9. It is easy to notice that the class of CF is a linear space with ordinary addition of two functions and ordinary multiplication of a function by a real number. However, it has an

We shall demonstrate that the notion of a Hilbert space is very useful in the study of immunization theory. It was named after a German mathematician David Hilbert (1862–1943) who is recognized as one of the most influential and universal mathematicians of the nineteenth century and the first half of twentieth century. By definition, a Hilbert space is a linear space, say H, which is additionally equipped with so-called scalar product (a generalization of the scalar product of two vectors from R<sup>n</sup>) defined for any two of its elements (vectors) h<sup>1</sup> ∈ H and h<sup>2</sup> ∈ H.

tiwi ra tð Þ<sup>i</sup> ½ � because parameters ti and wi remain the same. Adding Eq. (13) holding for b(t) to

ci exp �s tð Þ<sup>i</sup> ti ½ �

tiwia tð Þ<sup>i</sup> , (13)

tiwi b tð Þþ<sup>i</sup> c tð Þ<sup>i</sup> ½ �: (15)

: (14)

Practical Finance Strategies in Immunization http://dx.doi.org/10.5772/intechopen.78719 103

One of two classes of admissible shifts studied in [14] was the class of polynomials.

$$a(t) = a\_0 + a\_1t + a\_2t^2 + at^3 + \dots + at^{n-1} = \sum\_{j=1}^{j=n} a\_{j-1}t^{j-1}, \ t \in [t\_0; \mathbb{T}].\tag{11}$$

The new term structure was assumed to be of the form:

$$s^\*(0, t) = s(t) + \lambda a(t), t \in [t\_0, T]. \tag{12}$$

with a tð Þ satisfying Formula 11.

Definition 1. (see also [11], p. 859). A set S is said to be a linear space if the sum of its arbitrary 2 elements a∈ S and b∈ S belongs to S (a þ b∈S), and for any real number r the product of r and any element a∈ S belongs to S as well.

The most well-known linear spaces are probably the set of all real numbers R, a twodimensional Cartesian plane R<sup>2</sup> , a three-dimensional linear space R<sup>3</sup> , and their generalizations Rn, known as an n-dimensional linear spaces.

Definition 2. A set of k vectors v<sup>1</sup>, v<sup>2</sup>, …, v<sup>k</sup> is called linearly independent if each linear combination of these vectors λ<sup>1</sup> <sup>v</sup><sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> <sup>v</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>λ</sup><sup>k</sup> vk is a vector different from vector 0; see Definition 2.2 (p. 860).

Definition 3. A set of linearly independent vectors from a linear space S is called a base for S if each vector a∈ S is a linear combination of theirs, and this property does not hold any longer after removal of any of these base vectors.

All bases have the same size and there are many of them in each linear space S. Rn is a linear space with a natural addition x þ y ¼ x<sup>1</sup> þ y1; x<sup>2</sup> þ y2; ;…; xn þ yn � � of two vectors, and natural multiplication <sup>r</sup> � <sup>x</sup> <sup>¼</sup> ð Þ rx1;rx2;rx3;…;rxn <sup>∈</sup>R<sup>n</sup> of a vector <sup>x</sup> <sup>¼</sup> ð Þ <sup>x</sup>1; <sup>x</sup>2; ;…; xn by a real number <sup>r</sup>. The most popular base in <sup>R</sup><sup>n</sup> is the set of <sup>n</sup> vectors: <sup>v</sup><sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>0</sup>; <sup>0</sup>;…; <sup>0</sup> , <sup>v</sup><sup>2</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>1</sup>; <sup>0</sup>;…; <sup>0</sup> , <sup>v</sup><sup>3</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup>; <sup>1</sup>; <sup>0</sup>;…; <sup>0</sup> , and so on until vn <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup>; <sup>0</sup>;…0; <sup>1</sup> . Each vector, for example, (2, 3, �7, 10), is the following linear combination of the above base vectors: 2(1, 0, 0, 0) + 3(0, 1, 0, 0) + (�7)(0, 0, 1, 0) + 10(0, 0, 0, 1).

Remark 3. The below Formula (13), being a counterpart of Formula 8, gives a necessary and sufficient condition for immunization against shifts a(t) in case when a bond portfolio BP generates payments ck at instances t1, t2, t3, …, tm:

Practical Finance Strategies in Immunization http://dx.doi.org/10.5772/intechopen.78719

$$a(q)q = \sum\_{i=1}^{i=m} t\_i w\_i a(t\_i),\tag{13}$$

with weights

VA <sup>¼</sup> <sup>X</sup> k¼m

a tðÞ¼ a<sup>0</sup> þ a1t þ a2t

102 Immunization - Vaccine Adjuvant Delivery System and Strategies

with a tð Þ satisfying Formula 11.

dimensional Cartesian plane R<sup>2</sup>

combination of these vectors λ<sup>1</sup>

0) + (�7)(0, 0, 1, 0) + 10(0, 0, 0, 1).

generates payments ck at instances t1, t2, t3, …, tm:

Definition 2.2 (p. 860).

and any element a∈ S belongs to S as well.

Rn, known as an n-dimensional linear spaces.

after removal of any of these base vectors.

The new term structure was assumed to be of the form:

s

k¼1

One of two classes of admissible shifts studied in [14] was the class of polynomials.

<sup>2</sup> <sup>þ</sup> at<sup>3</sup> <sup>þ</sup> … <sup>þ</sup> atn�<sup>1</sup> <sup>¼</sup> <sup>X</sup>

Definition 1. (see also [11], p. 859). A set S is said to be a linear space if the sum of its arbitrary 2 elements a∈ S and b∈ S belongs to S (a þ b∈S), and for any real number r the product of r

The most well-known linear spaces are probably the set of all real numbers R, a two-

Definition 2. A set of k vectors v<sup>1</sup>, v<sup>2</sup>, …, v<sup>k</sup> is called linearly independent if each linear

Definition 3. A set of linearly independent vectors from a linear space S is called a base for S if each vector a∈ S is a linear combination of theirs, and this property does not hold any longer

All bases have the same size and there are many of them in each linear space S. Rn is a linear

multiplication <sup>r</sup> � <sup>x</sup> <sup>¼</sup> ð Þ rx1;rx2;rx3;…;rxn <sup>∈</sup>R<sup>n</sup> of a vector <sup>x</sup> <sup>¼</sup> ð Þ <sup>x</sup>1; <sup>x</sup>2; ;…; xn by a real number <sup>r</sup>. The most popular base in <sup>R</sup><sup>n</sup> is the set of <sup>n</sup> vectors: <sup>v</sup><sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>0</sup>; <sup>0</sup>;…; <sup>0</sup> , <sup>v</sup><sup>2</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>1</sup>; <sup>0</sup>;…; <sup>0</sup> , <sup>v</sup><sup>3</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup>; <sup>1</sup>; <sup>0</sup>;…; <sup>0</sup> , and so on until vn <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup>; <sup>0</sup>;…0; <sup>1</sup> . Each vector, for example, (2, 3, �7, 10), is the following linear combination of the above base vectors: 2(1, 0, 0, 0) + 3(0, 1, 0,

Remark 3. The below Formula (13), being a counterpart of Formula 8, gives a necessary and sufficient condition for immunization against shifts a(t) in case when a bond portfolio BP

<sup>v</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>λ</sup><sup>k</sup>

<sup>v</sup><sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

space with a natural addition x þ y ¼ x<sup>1</sup> þ y1; x<sup>2</sup> þ y2; ;…; xn þ yn

, a three-dimensional linear space R<sup>3</sup>

j¼n

j¼1

aj�<sup>1</sup>t j�1

<sup>∗</sup>ð Þ¼ <sup>0</sup>; <sup>t</sup> s tð Þþ <sup>λ</sup>a tð Þ, t∈½ � <sup>t</sup>0; <sup>T</sup> , (12)

ci exp �c tð Þ<sup>k</sup> tk ½ �: (10)

, t∈ ½ � t0; T : (11)

, and their generalizations

vk is a vector different from vector 0; see

� � of two vectors, and natural

$$w\_k = \frac{c\_k \exp\left[-s(t\_k)t\_k\right]}{\sum\_{i=1}^n c\_i \exp\left[-s(t\_i)t\_i\right]}.\tag{14}$$

Remark 4. The class of polynomials of the form (11), with fixed n, is a linear space. What is more, the subset of these polynomials satisfying Eq. (13) is a linear space, too.

Proof. Assume that a(t) satisfies Eq. (13). For any real number r, Eq. (13) implies ½ � ra qð Þ q ¼ i P¼m i¼1 tiwi ra tð Þ<sup>i</sup> ½ � because parameters ti and wi remain the same. Adding Eq. (13) holding for b(t) to Eq. (13) holding for c(t), one immediately obtains the required relationship.

$$[b(\boldsymbol{\eta}) + \boldsymbol{c}(\boldsymbol{\eta})]\boldsymbol{\eta} = \sum\_{i=1}^{i=m} t\_i \boldsymbol{w}\_i [b(t\_i) + \boldsymbol{c}(t\_i)].\tag{15}$$

Theorem 2. (see [11], Theorem 2.1). Let q denote the date when the single liability of L dollars has to be discharged by means of the cumulative value of assets (9) despite additive adverse shifts (11) (n is fixed) of interest rates s(t) so that the new interest rates will be of the form (12). Then, the subclass of shifts (11) for which immunization is secured is a (n�1)-dimensional linear space, denote it by IMMU, of the space of all polynomials (11), which itself has dimension n.

How to determine IMMU is demonstrated in [11] in pp. 858–860.

#### 3.2. Continuous functions are admissible shifts: a Hilbert space approach

The other class of admissible shifts studied in [11] was the class of all continuous functions (CF) defined as always on interval ½ � t0; T . As previously, the new interest rates (after a shift) satisfy Eq. (12) with a(t) standing this time for any CF. As previously, assets A(t) are given by Formula 9. It is easy to notice that the class of CF is a linear space with ordinary addition of two functions and ordinary multiplication of a function by a real number. However, it has an infinite number of independent vectors!

We shall demonstrate that the notion of a Hilbert space is very useful in the study of immunization theory. It was named after a German mathematician David Hilbert (1862–1943) who is recognized as one of the most influential and universal mathematicians of the nineteenth century and the first half of twentieth century. By definition, a Hilbert space is a linear space, say H, which is additionally equipped with so-called scalar product (a generalization of the scalar product of two vectors from R<sup>n</sup>) defined for any two of its elements (vectors) h<sup>1</sup> ∈ H and h<sup>2</sup> ∈ H.

103

A specific Hilbert space H\* of all CF (shifts) defined on interval ½ � t1; T was introduced in [11] and it was demonstrated that H\* had dimension m. However, the shifts of interest rates should be considered on interval ½ � t0; T because a random and unexpected shift a(t) might appear instantly after the acquisition of BP. In such a case, the dimension of H\* would be (m + 1), which is really the case. So, in this chapter, we correct and simplify the definition of a scalar product of two arbitrary continuous functions (shifts) f(t) and g(t), by letting

$$ = \sum\_{k=0}^{k=m} f(\mathbf{t}\_k) \mathbf{g}(\mathbf{t}\_k). \tag{16}$$

½a0P0ð Þþ q a1P1ð Þþ q a2P2ð Þþ q … þ amPmð Þq � � q �

cktk exp �s tð Þ<sup>k</sup> tk ½ �

It is worth to notice that after determination of polynomials Pkð Þt , 0 ≤ k ≤ m, all numbers P0ð Þq , P1ð Þq , P2ð Þq , …, Pmð Þq are known, as well as parameters q, ck, tk,s tð Þ<sup>k</sup> , so that a0, a1, a2, …, am remain the only unknown variables. The readers interesting in identifying subspace

3.3. Identification of continuous shifts against which a bond portfolio is immunized: the

A strict definition of triangular functions is given by Eqs. (20)–(23) below. Roughly speaking, a triangular function (sometimes called a tent function, or a hat function) is a function whose graph takes the shape of a triangle. Among our (m + 1) tent functions employed in this chapter, (m � 1) are isosceles triangles with height 1 and base 2, while the other two are perpendicular triangles with height 1 and base 1. Triangular functions have been successfully employed in signal processing as representations of idealized signals from which more realistic signals can

They also have applications in pulse code modulation as a pulse shape for transmitting digital signals, and as a matched filter for receiving the signals. Triangular functions are used to define the so-called triangular window, also known as the Bartlett window. Since they occur in the formula for Lagrange polynomials used in numerical analysis for polynomial interpolation, they are also called Lagrange functions. Their other applications include the Newton-Cotes

In the financial context, tent functions were employed in [17] for modeling shifts of the term structure of interest rates. The framework and assumptions made in this section are the same as in Section 3.2. Our purpose is to characterize the subspace IMMU of the Hilbert space H\* by

In this section, t1, t2, t3, …, tm ¼ T comprise not only all instances when a given bond portfolio BP generates payments, but also additionally the date q when the liability to pay L dollars has to be discharged. Below, we define m+1 triangular functions S0ð Þt , S1ð Þt , S2ð Þt , …, Smð Þt whose graphs are triangles with bases ½ � t0; t<sup>1</sup> , t½ � <sup>0</sup>; t<sup>2</sup> , t½ � <sup>1</sup>; t<sup>3</sup> , …, t½ � <sup>m</sup>�<sup>2</sup>; tm , t½ � <sup>m</sup>�<sup>1</sup>; tm . The first one S0ð Þt and the last one Smð Þt represent perpendicular triangles, while the remaining ones are isosceles triangles.

, t ∈½ � t0; t<sup>1</sup> and S0ðÞ¼ t 0 for t∈ ½ � t1; tm , (20)

, t∈ ½ � tm�<sup>1</sup>; tm and SmðÞ¼ t 0 for t∈ ½ � t0; tm�<sup>1</sup> , (21)

method of numerical integration, and Shamir's secret sharing scheme in cryptography.

� X k¼m

IMMU are referred to Example 4.2 (pp. 863–864).

be derived, for example, in kernel density estimation.

means of triangular functions based on results presented in [18].

t � t<sup>1</sup> t<sup>0</sup> � t<sup>1</sup>

t � tm�<sup>1</sup> tm � tm�<sup>1</sup>

S0ðÞ¼ t

SmðÞ¼ t

triangular functions approach

k¼1

X k¼m

ck exp �s tð Þ<sup>k</sup> tk ½ �¼ a0P0ð Þq

Practical Finance Strategies in Immunization http://dx.doi.org/10.5772/intechopen.78719

(19)

105

k¼1

It is good to know that in each Hilbert space H, one can measure a distance between any two elements <sup>h</sup><sup>1</sup> <sup>∈</sup> <sup>H</sup> and <sup>h</sup><sup>2</sup> <sup>∈</sup> <sup>H</sup> by the formula <sup>k</sup>h<sup>1</sup> � <sup>h</sup>2k, where <sup>k</sup>hk ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>&</sup>lt; h, h <sup>&</sup>gt; <sup>p</sup> is said to be a norm of vector h. In space H\*, the norm is therefore defined as follows:

$$\|h\| = \sqrt{} = \sqrt{\sum\_{k=0}^{k=m} h(t\_k)h(t\_k)}.\tag{17}$$

Clearly, khk ¼ 0 if and only if h tð Þ¼ <sup>k</sup> 0, 0 ≤ k ≤ m. A function h(t) belonging to H\* is treated as an element (vector) 0 (zero) if and only if khk ¼ 0. Therefore, kh<sup>1</sup> � h2k ¼ 0 means, then the two functions h1ð Þt and h2ð Þt are viewed as same on interval ½ � t0; T . It holds if and only if they coincide at all instances tk, 0 ≤ k ≤ m, when bond portfolio BP is paying cash.

In Theorem 3 to follow, we identify a base for H\* among polynomials. This approach is rather complicated since it involves the use of Gram-Schmidt orthogonalization procedure to determine base polynomials. In Section 3.3, a far more straightforward and easier to implement approach is presented where there is no need to identify base functions (shifts) because they are already given by Formulas (20)–(23).

Theorem 3. (compare Theorem 3.1 in [11]). Suppose a bond portfolio BP has been bought, and admissible shifts a(t) of a term structure s(t) are allowed to be continuous functions on interval ½ � t0; T . Then, the set of these shifts equipped with the scalar product (16) is an (m + 1)-dimensional Hilbert space H\*, where m is the number of instances when portfolio BP generates cash. The subset of these shifts, say IMMU, against which a holder of BP is immune (will be able to discharge the liability of L dollars to be paid at time q∈½ � t0; T by means of the cumulative value of assets (9)) is an m-dimensional subspace (depending to a large extent on BP) of the form

$$a(t) = a\_0 P\_0(t) + a\_1 P\_1(t) + a\_2 P\_2(t) + \dots + a\_m P\_m(t) \tag{18}$$

where the m+1 polynomials Pkð Þt , 0 ≤ k ≤ m, constitute a base of space H\*. This base may be determined by the Gram-Schmidt orthogonalization procedure, while the coefficients a0, a1, a2, …, am can be identified as solutions to the linear equation

$$\begin{aligned} \left[a\_0 P\_0(q) + a\_1 P\_1(q) + a\_2 P\_2(q) + \dots + a\_m P\_m(q)\right] \cdot q \cdot \sum\_{k=1}^{k=m} c\_k \exp\left[-s(t\_k)t\_k\right] &= \quad a\_0 P\_0(q) \\ \cdot \sum\_{k=1}^{k=m} c\_k t\_k \exp\left[-s(t\_k)t\_k\right] \end{aligned} \tag{19}$$

It is worth to notice that after determination of polynomials Pkð Þt , 0 ≤ k ≤ m, all numbers P0ð Þq , P1ð Þq , P2ð Þq , …, Pmð Þq are known, as well as parameters q, ck, tk,s tð Þ<sup>k</sup> , so that a0, a1, a2, …, am remain the only unknown variables. The readers interesting in identifying subspace IMMU are referred to Example 4.2 (pp. 863–864).

A specific Hilbert space H\* of all CF (shifts) defined on interval ½ � t1; T was introduced in [11] and it was demonstrated that H\* had dimension m. However, the shifts of interest rates should be considered on interval ½ � t0; T because a random and unexpected shift a(t) might appear instantly after the acquisition of BP. In such a case, the dimension of H\* would be (m + 1), which is really the case. So, in this chapter, we correct and simplify the definition of a scalar

k¼m

f tð Þ<sup>k</sup> g tð Þ<sup>k</sup> : (16)

vuut : (17)

<sup>&</sup>lt; h, h <sup>&</sup>gt; <sup>p</sup> is said to be a

k¼0

It is good to know that in each Hilbert space H, one can measure a distance between any two

Clearly, khk ¼ 0 if and only if h tð Þ¼ <sup>k</sup> 0, 0 ≤ k ≤ m. A function h(t) belonging to H\* is treated as an element (vector) 0 (zero) if and only if khk ¼ 0. Therefore, kh<sup>1</sup> � h2k ¼ 0 means, then the two functions h1ð Þt and h2ð Þt are viewed as same on interval ½ � t0; T . It holds if and only if they

In Theorem 3 to follow, we identify a base for H\* among polynomials. This approach is rather complicated since it involves the use of Gram-Schmidt orthogonalization procedure to determine base polynomials. In Section 3.3, a far more straightforward and easier to implement approach is presented where there is no need to identify base functions (shifts) because they

Theorem 3. (compare Theorem 3.1 in [11]). Suppose a bond portfolio BP has been bought, and admissible shifts a(t) of a term structure s(t) are allowed to be continuous functions on interval ½ � t0; T . Then, the set of these shifts equipped with the scalar product (16) is an (m + 1)-dimensional Hilbert space H\*, where m is the number of instances when portfolio BP generates cash. The subset of these shifts, say IMMU, against which a holder of BP is immune (will be able to discharge the liability of L dollars to be paid at time q∈½ � t0; T by means of the cumulative value of assets (9)) is an m-dimensional subspace (depending to a large extent on BP) of the form

where the m+1 polynomials Pkð Þt , 0 ≤ k ≤ m, constitute a base of space H\*. This base may be determined by the Gram-Schmidt orthogonalization procedure, while the coefficients a0, a1,

a2, …, am can be identified as solutions to the linear equation

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h tð Þ<sup>k</sup> h tð Þ<sup>k</sup>

a tðÞ¼ a0P0ð Þþ t a1P1ð Þþ t a2P2ð Þþ t … þ amPmð Þt (18)

X k¼m

k¼0

product of two arbitrary continuous functions (shifts) f(t) and g(t), by letting

104 Immunization - Vaccine Adjuvant Delivery System and Strategies

<sup>&</sup>lt; f,g <sup>&</sup>gt;<sup>¼</sup> <sup>X</sup>

elements <sup>h</sup><sup>1</sup> <sup>∈</sup> <sup>H</sup> and <sup>h</sup><sup>2</sup> <sup>∈</sup> <sup>H</sup> by the formula <sup>k</sup>h<sup>1</sup> � <sup>h</sup>2k, where <sup>k</sup>hk ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>k</sup>hk ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>&</sup>lt; h, h <sup>&</sup>gt; <sup>p</sup> <sup>¼</sup>

coincide at all instances tk, 0 ≤ k ≤ m, when bond portfolio BP is paying cash.

are already given by Formulas (20)–(23).

norm of vector h. In space H\*, the norm is therefore defined as follows:

#### 3.3. Identification of continuous shifts against which a bond portfolio is immunized: the triangular functions approach

A strict definition of triangular functions is given by Eqs. (20)–(23) below. Roughly speaking, a triangular function (sometimes called a tent function, or a hat function) is a function whose graph takes the shape of a triangle. Among our (m + 1) tent functions employed in this chapter, (m � 1) are isosceles triangles with height 1 and base 2, while the other two are perpendicular triangles with height 1 and base 1. Triangular functions have been successfully employed in signal processing as representations of idealized signals from which more realistic signals can be derived, for example, in kernel density estimation.

They also have applications in pulse code modulation as a pulse shape for transmitting digital signals, and as a matched filter for receiving the signals. Triangular functions are used to define the so-called triangular window, also known as the Bartlett window. Since they occur in the formula for Lagrange polynomials used in numerical analysis for polynomial interpolation, they are also called Lagrange functions. Their other applications include the Newton-Cotes method of numerical integration, and Shamir's secret sharing scheme in cryptography.

In the financial context, tent functions were employed in [17] for modeling shifts of the term structure of interest rates. The framework and assumptions made in this section are the same as in Section 3.2. Our purpose is to characterize the subspace IMMU of the Hilbert space H\* by means of triangular functions based on results presented in [18].

In this section, t1, t2, t3, …, tm ¼ T comprise not only all instances when a given bond portfolio BP generates payments, but also additionally the date q when the liability to pay L dollars has to be discharged. Below, we define m+1 triangular functions S0ð Þt , S1ð Þt , S2ð Þt , …, Smð Þt whose graphs are triangles with bases ½ � t0; t<sup>1</sup> , t½ � <sup>0</sup>; t<sup>2</sup> , t½ � <sup>1</sup>; t<sup>3</sup> , …, t½ � <sup>m</sup>�<sup>2</sup>; tm , t½ � <sup>m</sup>�<sup>1</sup>; tm . The first one S0ð Þt and the last one Smð Þt represent perpendicular triangles, while the remaining ones are isosceles triangles.

$$S\_0(t) = \frac{t - t\_1}{t\_0 - t\_1}, t \in [t\_0; t\_1] \text{ and } S\_0(t) = 0 \text{ for } t \in [t\_1; t\_m]. \tag{20}$$

$$S\_m(t) = \frac{t - t\_{m-1}}{t\_m - t\_{m-1}}, t \in [t\_{m-1}; t\_m] \text{ and } S\_m(t) = 0 \text{ for } t \in [t\_0; t\_{m-1}].\tag{21}$$

$$S\_k(t) = \frac{t - t\_{k-1}}{t\_k - t\_{k-1}}, t \in [t\_{k-1}; t\_k], 1 \le k \le m - 1,\tag{22}$$

For each vector <sup>v</sup> <sup>¼</sup> ð Þ <sup>v</sup>1; <sup>v</sup>2;…; vm <sup>∈</sup> Rm, the class Kv of such continuous shifts a(t) for which (25) holds was defined in [19]. When vector <sup>v</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>1</sup>;…; <sup>1</sup> <sup>∈</sup> Rm is used, then the corresponding class Kv comprises all parallel shifts for which a(t) = constant. We shall call

ing at time tk, the dedicated duration Dvð Þ¼ Bk tk � vk since all weights wi, except for wk, are

Theorem 5. The immunization of a bond portfolio BP against shifts a(t) from class Kv is

value of BP. Suppose that immediately after purchasing BP, interest rates s(t) will shift to new

details, see [19], p. 105. It is easy to notice that convexity of a zero-coupon bearing bond

unanticipating rate of return resulting from a shift a(t) of interest rates sð Þ� is given by the

where lim O(a) = 0 when a ! 0. Taking into account that a tð Þ<sup>k</sup> are small numbers of order 0.1% = 0.001, one concludes that the third term in (26) is really very small. Since each immunized bond portfolio BP satisfies Dvð Þ¼ BP q, the maximal unanticipating rate of return among immunized portfolios will be achieved when dedicated convexity Cvð Þ BP will be as high as

Assumption 1. All zero-coupon bearing bonds Bk, which form a bond portfolio BP and

2 tk 2vk

PV s½ � ð Þ� ¼ �Dvð Þ BP a qð Þþ Cvð Þ BP <sup>a</sup><sup>2</sup>

mature at tk, have mutually different dedicated durations, that is, Dv Bj

k¼m

k¼1

tiwivi:

i P¼m i¼1

PV s½ �¼ ðÞþ� <sup>a</sup>ð Þ� <sup>X</sup>

With s(t) standing for the current interest rates, PV s½ �¼ ð Þ� <sup>k</sup>

tiwivi the dedicated (for class Kv) duration. For zero-coupon bearing bond Bk, matur-

P¼m k¼1

ck exp �s tð Þ� <sup>k</sup> a tð Þ<sup>k</sup> ½ �tk:

<sup>2</sup> and call it dedicated (for class Kv) convexity of portfolio BP; for more

2. It was proved in [19], p. 106, that so-called

Oa tð Þ<sup>k</sup> ½ �a tð Þ<sup>k</sup>

� � 6¼ Dvð Þ Bn if and only

<sup>2</sup> (26)

ð Þþ <sup>q</sup> <sup>X</sup> k¼m

k¼1

ck exp �s tð Þ<sup>k</sup> tk ½ � is the present

Practical Finance Strategies in Immunization http://dx.doi.org/10.5772/intechopen.78719 107

Dv <sup>¼</sup> <sup>i</sup> P¼m i¼1

equal to 0.

secured if and only if q ¼ Dvð Þ¼ BP

levels s\*(t) = s(t) + a(t). Then

4.1. Convexity of a bond portfolio

maturing at tk is given by the formula Cv <sup>¼</sup> <sup>1</sup>

PV s½ �� ðÞþ� að Þ� PV s½ � ð Þ�

if j 6¼ n, that is, tj � vj 6¼ tn � vn \$ j 6¼ n.

2 Pm k¼1 tk <sup>2</sup>wkvk

Set Cvð Þ¼ BP <sup>1</sup>

formula:

possible.

$$S\_k(t) = \frac{t - t\_{k+1}}{t\_k - t\_{k+1}}, t \in [t\_k; t\_{k+1}];\\ S\_k(t) = 0 \text{ elsewhere in } [t\_0; T]. \tag{23}$$

The following result is well known.

Remark 5. Each continuous function a(t) defined on ½ � t0; T attains the same values as the function b tðÞ¼ að Þ� 0 S0ð Þþ t a tð Þ� <sup>1</sup> S1ð Þþ t a tð Þ� <sup>2</sup> S2ð Þþ t … þ a tð Þ� <sup>m</sup> Smð Þt (built up with (m + 1) triangular functions) at all points tk, 0 ≤ k ≤ m. Therefore, a(t) may be identified in H\* with the piecewise linear function b(t) because the distance between a(t) and b(t) in H\* is zero: ||b(t) – a (t)|| = 0.

It is a nice exercise to prove the following result.

Remark 6. The Lagrange functions S0ð Þt , S1ð Þt , S2ð Þt , …, Smð Þt given by (20)–(23) constitute a base for Hilbert space H\* of all admissible (continuous) shifts defined on ½ � t0; T .

Theorem 4. The set IMMU of all shifts (continuous functions) against which a bond portfolio BP with payouts represented by (9) and the new term structure given by (12) is immunized constitutes an m-dimensional linear subspace in the (m + 1)-dimensional Hilbert space H\*.

Two examples illustrating how to identify IMMU are worked out in detail in [18], pp. 531–537. A special attention is given to continuity properties of subspace IMMU; see [18], pp. 534–537.
